Get angle of a face relative to object origin?
The way to get the information I want per-face out is to run:
euler_rotation = matrix_rotate.to_euler()
euler_rotation

for the Euler rotation, and
matrix_translation

for the translation, taking only the magnitude portion of the matrix's x, y, and z rows. This is as a Euler transformation.  To do it as a Quaternion, I can simply run:
my_quaternion = matrix_translation @ matrix_rotate.to_4x4()
my_quaternion

and get a 4-number output that I believe includes both position and rotation transformations for the joint.